Q.Describe the displacement method for determining the focal length of a convex lens using an object and a fixed screen, define the term conjugate points, and derive the formula $f=\dfrac{D^{2}-d^{2}}{4D}$.

With the object and screen fixed a distance $D>4f$ apart, a convex lens slid along the line between them produces a sharp image on the screen at two different lens positions: one where the lens sits closer to the object (giving a magnified real image) and one where it sits closer to the screen (giving a diminished real image). Because the thin lens formula treats the object and image distances symmetrically, these two positions are conjugate — the object distance in one position equals the image distance in the other. Writing $d$ for the distance between the two lens positions, the object distance in the first position is $u_1=-\dfrac{D-d}{2}$ (and correspondingly $v_1=-(D-u_1)=-\dfrac{D+d}{2}$, taking magnitudes appropriately for the geometry). Substituting into the lens formula $\dfrac1{v_1}-\dfrac1{u_1}=\dfrac1f$ and simplifying algebraically (the two positions give equations that are mirror images of each other in $u$ and $v$) yields, after clearing fractions, $f=\dfrac{D^2-d^2}{4D}$. Because this result depends only on $D$ and $d$ — both di …

Set up the two lens positions as object/image-distance pairs that are exact mirror images of each other in terms of $D$ and …

• Forgetting the requirement $D>4f$, without which no two real conjugate positions exist at all.
• Trying to use only one lens position (that is simply the ordinary lens-formula method, not the displacement method). …